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Reflectionless Measures and the Mattila-Melnikov-Verdera Uniform Rectifiability Theorem

Part of the Lecture Notes in Mathematics book series (LNM,volume 2116)

Abstract

The aim of these notes is to provide a new proof of the Mattila-Melnikov-Verdera theorem on the uniform rectifiability of an Ahlfors-David regular measure whose associated Cauchy transform operator is bounded. They are based on lectures given by the second author in the analysis seminars at Kent State University and Tel-Aviv University.

Keywords

  • Uniform Rectifiability
  • Reflectionless
  • Cauchy Transform
  • Dyadic Square
  • Tangible Measures

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Notes

  1. 1.

    In the notation of the previous section, \(\Gamma = \Gamma _{\ell_{0}}{\bigl ( \tfrac{\ell(Q)} {2} \bigl )}\), for some \(\ell_{0} \leq \tfrac{\ell(Q)} {2}\).

References

  1. G. David, S. Semmes, Analysis of and on Uniformly Rectifiable Sets. Mathematical Surveys and Monographs, vol. 38 (American Mathematical Society, Providence, 1993)

    Google Scholar 

  2. P.W. Jones, Rectifiable sets and the traveling salesman problem. Invent. Math. 102(1), 1–15 (1990)

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. J.C. Léger, Menger curvature and rectifiability. Ann. Math. 149(3), 831–869 (1999)

    CrossRef  MATH  Google Scholar 

  4. P. Mattila, Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability. Cambridge Studies in Advanced Mathematics, vol. 44 (Cambridge University Press, Cambridge, 1995)

    Google Scholar 

  5. P. Mattila, Cauchy singular integrals and rectifiability in measures of the plane. Adv. Math. 115(1), 1–34 (1995)

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. P. Mattila, D. Preiss, Rectifiable measures in \(\mathbb{R}^{n}\) and existence of principal values for singular integrals. J. Lond. Math. Soc. (2) 52(3), 482–496 (1995)

    Google Scholar 

  7. P. Mattila, J. Verdera, Convergence of singular integrals with general measures. J. Eur. Math. Soc. 11(2), 257–271 (2009)

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. P. Mattila, M. Melnikov, J. Verdera, The Cauchy integral, analytic capacity, and uniform rectifiability. Ann. Math. 144, 127–136 (1996)

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. M. Melnikov, A. Poltoratski, A. Volberg, Uniqueness theorems for Cauchy integrals. Publ. Mat. 52(2), 289–314 (2008)

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. F. Nazarov, X. Tolsa, A. Volberg, On the uniform rectifiability of AD regular measures with bounded Riesz transform operator: the case of codimension 1. Preprint (2012). arXiv:1212.5229

    Google Scholar 

  11. D. Preiss, Geometry of measures in \(\mathbb{R}^{n}\): distribution, rectifiability, and densities. Ann. Math. (2) 125(3), 537–643 (1987)

    Google Scholar 

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Correspondence to Fedor Nazarov .

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Jaye, B., Nazarov, F. (2014). Reflectionless Measures and the Mattila-Melnikov-Verdera Uniform Rectifiability Theorem. In: Klartag, B., Milman, E. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2116. Springer, Cham. https://doi.org/10.1007/978-3-319-09477-9_15

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